Freudenthal triple systems via root system methods Público

Abstract

For a Lie algebra 𝔤 of type B, D, E or F, we can apply a grading 𝔤 = 𝔤−2⊕𝔤−1⊕𝔤0⊕𝔤1⊕𝔤2 and then define a quartic form and a skew-symmetric bilinear form on 𝔤1, thereby constructing a Freudenthal triple system. The structure of the Freudenthal triple system is examined using root system methods available in the Lie algebra context. In the important cases 𝔤= E8 and 𝔤 = D4, we determine the groups stabilizing the quartic form and both the quartic and bilinear forms.

Table of Contents

1 Introduction 1 1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Prior work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Technical background 8 2.1 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Classification of root systems . . . . . . . . . . . . . . . . . . 12 2.3 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Our situation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Roots of α-height 1 . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 General Results 27 3.1 The bilinear and quartic forms . . . . . . . . . . . . . . . . . . 27 3.2 Strictly regular elements . . . . . . . . . . . . . . . . . . . . . 34 3.3 Freudenthal triple systems . . . . . . . . . . . . . . . . . . . . 44 3.4 Computation of the 4-linear form . . . . . . . . . . . . . . . . 45 4 Special Results 51 4.1 Eigenspace decomposition of 𝔤1 . . . . . . . . . . . . . . . . . 51 4.2 Characterization of the orbits . . . . . . . . . . . . . . . . . . 53

4.3 Related groups . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 The stabilizer of the quartic form: G = E8 . . . . . . . . . . . 56 4.5 The stabilizer of the quartic form: G = D4 . . . . . . . . . . . 66 5 Conclusion 73 5.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Bibliography 77

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Palabra Clave	algebraic groups Lie algebras representation theory
Committee Chair / Thesis Advisor	Garibaldi, Skip, Emory University
Committee Members	Raman, Parimala, Emory University Brussel, Eric, Emory University

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